On the theoretical side, in [1]}, the authors propose a method to set edge weights that achieve the fastest mixing WRW for a given target stationary distribution.
This technique, although related, is not applicable in our context.
First, [1]} requires the knowledge of the graph, which makes it inapplicable to \(G\) ,
yet possibly feasible in \(G^C\)  (after estimating some limited information about \(G^C\)  as in Sec. REF ).
In the latter case, however, even given a perfect knowledge of \(G^C\) , [1]} often assigns weight 0 to some self-loops,
which likely makes the underlying graph \(G\)  disconnected.
Finally, and most importantly, [1]} takes a target stationary distribution as input. By taking \(\textrm {w}^{\scriptscriptstyle \textrm {WIS}}\) , we will face exactly the same problems of potentially poor convergence (Sec. REF ) and “black holes” (Sec. REF ) as we addressed by S-WRW.
